How is the charge transfered through a diaphragm in a galvanic/electrolytic cell?

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In basic chemistry courses you will learn that diaphragm prevents mixing of solutions between half-cells while maintaining ionic conductivity. What exactly carries charge through diaphragm if it's neither electrons, nor ions from the solution? What is the molecular mechanism behind that?
 

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Borek
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Ions do travel through the membrane. However, you usually design the system in such a way that traveling ions are those that will not react directly.
 
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Ions do travel through the membrane. However, you usually design the system in such a way that traveling ions are those that will not react directly.
So it's some kind of (double) ion exchanger?
 

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